Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 444 https://internationalpubls.com Homomorphism of Tripolar Fuzzy Soft Ternary Γ-Semiring E. Meera Prasad 1,2 , D. Madhusudhana Rao 3 , 4 G. Srinivasa Rao, G. Suresh Kumar 4 . M. Vasantha 5 1 Research Scholar, Department of Mathematics, KoneruLakshmaiah Education Foundation, Vaddeswaram, Guntur, A.P. India. Emai:meeraprasad.6499@gmail.com 2 Deputy Transport Commissioner, Guntur(Dt), Guntur, A.P., India. 3 Department of Mathematics, Government Degree College for Women(A), Guntur, A.P. India. Email:dmrmaths@gmail.com 4 Department of Mathematics & Statistics, VFSTR Deemed to be University, Vadlamudi, Guntur, A.P., India.Email:gsrinulakshmi77@gmail.com 4 Department of Mathematics, KoneruLakshmaiah Education Foundation, Vaddeswaram, Guntur, A.P. India. Email: drgsk006@kluniversity.in 5 Department of H&S, Mallareddy Engineering College for Women, Hyderabad, Telangana, India. Email: bezawadavasantha@gmail.com Article History: Received: 01-06-2024 Revised: 03-07-2024 Accepted: 29-07-2024 Abstract: The notions of a tripolar fuzzy soft ternary Γ−Semirings, a tripolar fuzzy soft ternary Γ−Semiring homomorphism and a tripolar fuzzy soft ideal in ternary Γ−Semirings are discussed, and related properties are investigated. On the other hand, in this paper, we also define the image and pre-image of tripolar fuzzy soft ternary Γ−Semirings. Some properties and results involving these concepts are stated and proved. Keywords: Soft set, fuzzy Soft set, tripolar fuzzy soft set, tripolar fuzzy soft ternary Γ−Semiring, tripolar fuzzy soft ideal, tripolar fuzzy soft ternary Γ−Semiring homomorphism. 1. Introduction In the year 2017 Revathi et.al [7, 8] introduced the concept of “fuzzy ideals” in “ternary gamma semirings”. In the year 2018, they [9,10,11] studied “fuzzy regular ternary gamma semirings”, “completely prime fuzzy ideals” and “prime fuzzy ideals in ternary gamma semirings”. In the year 2020 E. Meera Prasad et.al [6] introduced the notion of “fuzzy soft Bi-ideals” over “ternary gamma semirings”. Satish. T et. al [14] studied about “fuzzy soft ideals in ternary gamma semirings”. G. Srinivasa Rao et.al [ ] studies about ternary gamma semirings extensively. In this paper we introduce the notion “tripolar fuzzy soft ternary Γ-semiring homomorphism”. Throughout this paper we indicate “ternary gamma semiring” as TGSR, “Fuzzy soft ternary gamma Semiring” as FSTGSR, “tripolar fuzzy soft set” as TFSs, “Tripolar Fuzzy soft ternary gamma semiring” as TFSTGSR, “Tripolar fuzzy soft gamma ideal” as TFSI and “Tripolar fuzzy soft ternary Γ-semiring homomorphism” as TFSTGSRH, unless otherwise stated. 2. Preliminaries: Definition 2.1: A “tripolar fuzzy set” T in a universe set M is an object having the form T = {(a, νT(a), γT(a), βT(a)) | a ∈ M, 0 ≤ νT(a) + γT(a) ≤ 1}, where νT : M ⟶ [0, 1], γT : M ⟶ [0, 1] and βT : M ⟶ [-1, 0] such that 0 ≤ νT(a) + γT(a) ≤ 1. Since νT characterises the extent that the mailto:meeraprasad.6499@gmail.com mailto:dmrmaths@gmail.com mailto:drgsk006@kluniversity.in mailto:bezawadavasantha@gmail.com Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 445 https://internationalpubls.com element a satisfies the property corresponding the tripolar fuzzy set T, γT characterises the extent that the element a satisfies the irrelevant (non property) corresponding the tripolar fuzzy set T and βT characterises the extent that the element a satisfies to the implicit counter property of the tripolar fuzzy set T. We use the notion T = { νT, γT, βT} instead of T = {(a, νT(a), γT(a), βT(a)) | a ∈ M, 0 ≤ νT(a) + γT(a) ≤ 1}. Definition 2.2: A TFSs (δ, V, Γ) over TGSR M is nothing but TFSTGSR over TGSR M if ν(a) = {(p, νδ(a) (p), γ δ(a)(p), β δ(a)(p) / p ∈ M, a ∈ V}, where νδ(a): M → [0, 1], γ δ(a): M → [0, 1] and β δ(a): M → [-1, 0], ∋ 0 ≤ νδ(a) + γ δ(a) ≤ 1 and ∀ p ∈ M satisfying the following conditions: (1) νδ(a)(j + m) ≥ min { νδ(a)(j), νδ(a)(m)} (2) γ δ(a)(j + m) ≤ max { γ δ(a)(j), γ δ(a)(m)} (3) β δ(a)(j + m) ≤ max { β δ(a)(j), β δ(a)(m)} (4) νδ(a)(j𝛼m𝜃k) ≥ min { νδ(a)(j), νδ(a)(m), νδ(a)(k)} (5) γ δ(a)(j𝛼m𝜃k) ≤ max { γ δ(a)(j), γ δ(a)(m), γ δ(a)(k)} (6) β δ(a)(j𝛼m𝜃k) ≤ max { β δ(a)(j), β δ(a)(m), β δ(a)(k)} ∀ j, m, k ∈ M, a ∈ V, α, ζ ∈ Γ. Definition 2.3: A TFSs (δ, V, Γ) over TGSR M is nothing but TFSI over TGSR M if (1) νδ(a)(j + m) ≥ min { νδ(a)(j), νδ(a)(m)} (2) γ δ(a)(j + m) ≤ max { γ δ(a)(j), γ δ(a)(m)} (3) β δ(a)(j + m) ≤ max { β δ(a)(j), β δ(a)(m)} (4) νδ(a)(j𝛼m𝜃k) ≥ max { νδ(a)(j), νδ(a)(m), νδ(a)(k)} (5) γ δ(a)(j𝛼m𝜃k) ≤ min { γ δ(a)(j), γ δ(a)(m), γ δ(a)(k)} (6) β δ(a)(j𝛼m𝜃k) ≤ min { β δ(a)(j), β δ(a)(m), β δ(a)(k)} ∀ j, m, k ∈ M, a ∈ V, α, ζ ∈ Γ. Definition 2.4: If M1 and M2 are two TGSRs, a function π: M1 → M2 is called a homomorphism TGSR if π(j + m) = π(j) + π(m) and π(j𝛼m𝜃k) = π(j)απ(m)ζπ(k), ∀ j, m, k ∈ M1, α, 𝜃 ∈ Γ. Let M1 and M2 two sets and ψ : M1 ⟶ M2 is any function. A bipolar fuzzy subset ε of M1 is nothing but a ε-invariant ψ(p) = ψ(q) ⟹ ε(p) = ε(q). Let ψ: M1 ⟶ M2 is any function, ε = (ε + , ε - ) and ε = (ε + , ε - ) are bipolar fuzzy subsets in M1, M2 respectively. Then the image ψ(𝜀) of ε is the bipolar fuzzy subset ψ(ε) = ((ψ(ε)) + , (ψ(ε)) - ) of M2 defined as      max{ ( ) ( ): ( ): ( ) } 0 otherwise p p p if pp               max{ ( ) ( ): ( ): ( ) } 0 otherwise p p p if pp          And the pre-image ψ - (ε) of ε under ψ is the bipolar fuzzy subset of M1 defined by for p ∈ M1 ((ψ - 1 (ε)) + (p) = ε + (ψ(a)) and ((ψ -1 (ε)) -1 ( p) = ε -1 (ψ(a)). For more preliminaries refer the references and their references. For more preliminaries consider the references and their references. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 446 https://internationalpubls.com 3: Homomorphism in tripolar fuzzy soft Ternary Γ−semiring: In this section the homomorphism concept over TFSTGSR is introduced and studied their properties. Definition 3.1: Let (δ, V, Γ) and (ε, W, Γ) TFSs over TGSR M1 and M2 and ψ1: M1 ⟶ M2, ψ2 : V ⟶ W ware two functions such that V, W ware parameter sets for the crisp sets M1 and M2, then (ψ1, ψ2) is nothing but tripolar fuzzy soft function from M1 to M2. Definition 3.2: Let (δ, V, Γ) and (ε, W, Γ) TFSs over TGSR M1 and M2 and (ψ1, ψ2) is tripolar fuzzy soft function from M1 to M2. Then (ψ1, ψ2) is called Tripolar fuzzy soft ternary Γ-semiring homomorphism if satisfying the following laws: 1) Ψ1 is a ternary Γ-semiring homomorphism from M1 onto M2. 2) Ψ2 is a mapping from V onto W. 3) 21 ( ) ( )( )x x    , 21 ( ) ( )( )x x    and 21 ( ) ( )( )x x    Note 3.3: If there exist a TFSTGSRH between (δ, V, Γ) and (ε, W, Γ), then we say that (δ, V, Γ) is soft homomorphic to (ε, W, Γ). Definition 3.4: If (ψ1, ψ2) is a tripolar fuzzy soft function from M1 to M2. The pre image of (ε, W, Γ) under the tripolar fuzzy soft function (ψ1, ψ2) devoted by (ψ1, ψ2) -1 (ε, W, Γ) defined as (ψ1, ψ2) -1 (ε, W, Γ) = 1 1 1 2( ( ), ( ))W    is a TFSs. Theorem 3.5: If (ε, W, Γ) is a TFSTGSR over TGSR M2, ψ: M1 ⟶ M2 is a monomorphism and for each w ∈W, define (ψε)w(m) = ψw( ε(m)) for all m ∈ M1. Then (ψε, W, Γ) is a TFSTGSR over TGSR M2. Proof: Suppose j, m, k ∈ M, w ∈ W and α, 𝜃 ∈ Γ. Then 1. (ψε)w(j + m) = ψw(ε(j + m )) = νψw(ε(j )+ε(m)) ≥min{ νψw(ε(j )), νψw(ε(m))} = min {(ψε)w(j), (ψε)w(m)} 2. (ψε)w(j + m) = ψw(ε(j + m )) = γψw(ε(j )+ε(m)) ≤ max {γψw(ε(j )), γψw (ε(m))} = max {(ψε)w(j), (ψε)w(m)}. 3. (ψε)w(j + m) = ψw(ε(j + m )) = 𝛽ψw(ε(j )+ε(m)) ≤ max {βψw(ε(j )), βψw (ε(m))} = max {(ψε)w(j), (ψε)w(m)}. 4. (ψε)w(j𝛼m𝜃k) = ψw(ε(j𝛼m𝜃k)) = νψw(ε(j )𝛼ε(m)𝜃𝜀(k)) ≥min{ νψw(ε(j )), νψw(ε(m)), νψw(ε(k))} = min {(ψε)w(j), (ψε)w(m), (ψε)w(k)}. 5. (ψε)w(j𝛼m𝜃k) = ψw(ε(j𝛼m𝜃k)) = γψw(ε(j )𝛼ε(m)𝜃𝜀(k)) ≤max { γψw(ε(j )), γψw(ε(m)), γψw(ε(k))} = max {(ψε)w(j), (ψε)w(m), (ψε)w(k)}. 6. (ψε)w(j𝛼m𝜃k) = ψw(ε(j𝛼m𝜃k)) = βψw(ε(j )𝛼ε(m)𝜃𝜀(k)) ≤max { βψw(ε(j )), βψw(ε(m)), βψw(ε(k))} = max {(ψε)w(j), (ψε)w(m), (ψε)w(k)}. Therefore (ψε)w(m) is a tripolar fuzzy soft Γ−subsemiring of M. Thus (ψε, W, Γ) is a tripolar fuzzy soft ternary Γ−semiring over M2. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 447 https://internationalpubls.com Theorem 3.6: If (𝛗, W, 𝚪) TFTSR over TGSR M, 𝛘 is an endomorphism of M and defined (𝛗𝛘)w = 𝛗w𝛘 for each w ∈W. Then (𝛗𝛘, W, 𝚪) is a TFSTGSR over M. Proof: Suppose j, m, k ∈ M, w ∈ W and α, 𝜃 ∈ Γ. Then 1. (ν𝛘)w(j + m) = νφ(w)(χ(j + m))= νφ(w)(χ(j) + χ(m)) ≥min { νφ(w)(χ(j), νφ(w)(χ(m) } = min { ((νχ)w(j)), ((νχ)w(m))} 2. (γ𝛘)w(j + m) = γφ(w)(χ(j + m))= γφ(w)(χ(j) + χ(m)) ≤ max { γφ(w)(χ(j), γφ(w)(χ(m) } = max { ((γχ)w(j)), ((γχ)w(m))} 3. (𝛽𝛘)w(j + m) = 𝛽φ(w)(χ(j + m))= 𝛽φ(w)(χ(j) + χ(m)) ≤ max{ 𝛽φ(w)(χ(j), 𝛽φ(w)(χ(m) } = max { ((𝛽χ)w(j)), ((𝛽χ)w(m))} 4. (ν𝛘)w(j𝛼m𝜃k)= νφ(w)(χ(j𝛼m𝜃k))= νφ(w)(χ(j)𝛼χ(m)𝜃 χ(k)) ≥ min { νφ(w)(χ(j), νφ(w)(χ(m), νφ(w)(χ(k)} = min {(ν𝛘)w(j), (ν𝛘)w(m), (ν𝛘)w(k)} 5. (γ𝛘)w(j𝛼m𝜃k)= γφ(w)(χ(j𝛼m𝜃k))= γφ(w)(χ(j)𝛼χ(m)𝜃 χ(k)) ≤ max {γφ(w)(χ(j), γφ(w)(χ(m), γφ(w)(χ(k)} = max {(γ𝛘)w(j), (γ𝛘)w(m), (γ𝛘)w(k)}. 6. (𝛽𝛘)w(j𝛼m𝜃k)= 𝛽φ(w)(χ(j𝛼m𝜃k))= 𝛽φ(w)(χ(j)𝛼χ(m)𝜃 χ(k)) ≤ max {𝛽φ(w)(χ(j), 𝛽φ(w)(χ(m), 𝛽φ(w)(χ(k)} = max {(𝛽𝛘)w(j), (𝛽𝛘)w(m), (𝛽𝛘)w(k)}. Thus (𝛗𝛘)w is a TFGSR of M. Then (𝛗𝛘, W, 𝚪) is a TFSTGSR over M. Theorem 3.7: If φ : M1→ M2 is an epimorphism of TGSR and (ρ, W, Γ) is a “tripolar fuzzy soft right ideal” over M2. If for each w ∈ W, w = 1( )w  then (ζ, W, Γ) is a “tripolar fuzzy soft right ideal” over M1. Proof: If w ∈ W and α, ζ ∈ Γ. Then ρx is a “tripolar fuzzy soft right ideal” over M2. If j, m, k ∈ M1 and α, ζ ∈ Γ, then: 1. 1  ( νw(j + m)) = ν𝜌(w) (φ(j + m)) = ν𝜌(w)( φ(j) + φ(m)) ≥ min{ ν𝜌(w)( φ(j) , ν𝜌(w)( φ(m))} = min{ 1  (νw(j )), 1  (νw(m ))} 2. 1  (γw)( j + m) = γ𝜌(w) (φ(j + m)) = γ𝜌(w)( φ(j) + φ(m)) ≤ max{ γ𝜌(w)( φ(j)), γ𝜌(w)( φ(m))} = max{ 1  (γw)( j), 1  (γw)( m)}. 3. 1  (βw)( j + m) = β𝜌(w) (φ(j + m)) = β𝜌(w)( φ(j) + φ(m)) ≤ max{ β𝜌(w)( φ(j)), β𝜌(w)( φ(m))} Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 448 https://internationalpubls.com = max{ 1  (βw)( j), 1  (βw)( m)}. 4. 1  ( νw(j𝛼m𝜃k)) = ν𝜌(w) (φ(j𝛼m𝜃k)) = ν𝜌(w) ( φ(j)𝛼φ(m)𝜃 φ(k)) ≥ ν𝜌(w) (φ(j) = 1  ( νw(j)). 5. 1  ( γw(j𝛼m𝜃k)) = γ𝜌(w)(φ(j𝛼m𝜃k)) = γ𝜌(w)( φ(j)𝛼φ(m)𝜃 φ(k)) ≤ γ𝜌(w)( φ(j) = 1  ( γw(j)). 6. 1  (βw(j𝛼m𝜃k)) = β𝜌(w) (φ(j𝛼m𝜃k)) = β𝜌(w) ( φ(j)𝛼φ(m)𝜃 φ(k)) ≤ β𝜌(w) ( φ(j) = 1  (βw(j)). Therefore w = 1( )w  is a “tripolar fuzzy right rideal” of M1. Thus (δ, W, Γ) is a “tripolar fuzzy soft right ideal” over M1. Theorem 3.7 is also true for tripolar fuzzy left ideal. Proposition 3.8. If M1 and M2 are TGSRs, 𝛘 : M1 → M2 is a TGSRH and φ is a 𝛘−invariant bipolar fuzzy subset of M1, if w = 𝛘(u) then 𝛘(φ)(w) = φ(u); u ∈ M1. Proof: Straight forward. Theorem 3.9: If (𝛗, W, 𝚪) is a tripolar fuzzy soft right ideal over TGSR M1 and 𝛘 is a homomorphism from M1 onto M2. For each w ∈ W, 𝛗w is a 𝝌−invariant bipolar fuzzy right rideal of M1, if ζw = 𝛘(𝛗w) then (ζ, W, 𝚪) is a tripolar fuzzy soft right ideal over M2. Proof: Let m1, m2, m3 ∈ M2, w ∈ W, α, 𝛽 ∈ Γ. Then ∃ m4, m5, m6 ∈ M1 ∋ χ(m4) = m1, χ(m5) = m2, χ(m6) = m3, m1 + m2 = χ(m4 + m5) & m1𝛼m2𝛽m3 = χ(m4𝛼m5𝛽m6). 𝛗w is a 𝝌−invariant. Thus, by proposition 3.8, we have: 1. νς(w)(m1 + m2) = 𝛘(ν𝛗w)(m1 + m2) = ν𝛗w( m4 + m5) ≥ min {ν𝛗w(m4), ν𝛗w(m5)} = min {𝛘(ν𝛗w)(m1), 𝛘(ν𝛗w)(m2)} = min {νς(w) (m1), νς(w) (m1)} 2. γς(w)(m1 + m2) = 𝛘(γ𝛗w)(m1 + m2) = γ𝛗w( m4 + m5) ≤ max{γ𝛗w( m4), γ𝛗w( m5)} = max{𝛘(γ𝛗w)(m1), 𝛘(γ𝛗w)(m2)} = max{γς(w)(m1), γς(w)(m2)} 3. βς(w)(m1 + m2) = 𝛘(β𝛗w)(m1 + m2) = β𝛗w( m4 + m5) ≤ max{β𝛗w( m4), β𝛗w( m5)} = max{𝛘(β𝛗w)(m1), 𝛘(β𝛗w)(m2)} = max{βς(w)(m1), βς(w)(m2)} 4. νς(w)(m1𝛼m2𝛽m3) = 𝛘(ν𝛗w)(m1𝛼m2𝛽m3) = ν𝛗w(𝜒(m4𝛼m5𝛽m6)) = ν𝛗w(𝜒(m4)𝛼𝜒(m5)𝛽 𝜒(m6)) ≥ ν𝛗w(𝜒(m4) = 𝛘(ν𝛗w)(m1) = νς(w)( m1). 5. γς(w)(m1𝛼m2𝛽m3) = 𝛘(γ𝛗w)(m1𝛼m2𝛽m3) = γ𝛗w(𝜒(m4𝛼m5𝛽m6)) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 449 https://internationalpubls.com = γ𝛗w(𝜒(m4)𝛼𝜒(m5)𝛽 𝜒(m6)) ≤ γ𝛗w(𝜒(m4) = 𝛘(γ𝛗w)(m1) = γς(w)( m1). 6. βς(w)(m1𝛼m2𝛽m3) = 𝛘(β𝛗w)(m1𝛼m2𝛽m3) = β𝛗w(𝜒(m4𝛼m5𝛽m6)) = β𝛗w(𝜒(m4)𝛼𝜒(m5)𝛽𝜒(m6)) ≤ β𝛗w(𝜒(m4) = 𝛘(β𝛗w)(m1) = βς(w)( m1). then δw is a tripolar fuzzy ideal of M2. Hence (δ, W, 𝚪) is a tripolar fuzzy soft right ideal over M2. Theorem 3.10: If (𝛗1, W1, 𝚪) and (𝛗2, W2, 𝚪) are two bipolar FSTGSR over M1 and M2 respectively, and (𝛘, ψ) is a TFSTGSRH from (𝛗1, W1, 𝚪) onto (𝛗2, W2, 𝚪). Then (𝛘(𝛗1), M2, 𝚪) is a TFSTGSR over M2. Proof: By definition 3.2., 𝛘 is a TGSRH from M1 into M2 and ψ is a mapping from W1 into W2. For each w2 ∈W2 ∃ w1 ∈W1 ∋ ψ(w1) = w2. Define 2 ( ( ))w  = 1 ( )w  . If m4, m5, m6 ∈M2, 𝛼, 𝜃 ∈Γ. Then ∃ m1, m2, m3 ∈M1 ∋ χ(m1) = m4, χ(m2) = m5, χ(m3) = m6, χ(m1 + m2) = m4 + m5 & χ(m1𝛼m2𝜃m3) = m4𝛼m5𝜃m6. Thus, we get 1. 1 1( ) ( )( ( )) w   (m1 + m2) = 1( ) 1( ( ( ))w  (m1 + m2) = 1( ) 1( )w ( m4 + m5) ≥ min { 1( ) 1( )w (m4), 1( ) 1( )w (m5)} = min { 1( ) 1( ( ( ))w  (m1), 1( ) 1( ( ( ))w  (m2)} = min { 1 1( ) ( )( ( )) w   (m1), 1 1( ) ( )( ( )) w   (m2)} 2. 1 1( ) ( )( ( )) w  (m1 + m2) = 1( ) 1( ( ( ))w  (m1 + m2) = 1( ) 1( )w ( m4 + m5) ≤ max { 1( ) 1( )w (m4), 1( ) 1( )w (m5)} = max { 1( ) 1( ( ( ))w  (m1), 1( ) 1( ( ( ))w  (m2)} = max{ 1 1( ) ( )( ( )) w  (m1), 1 1( ) ( )( ( )) w  (m2)} 3. 1 1( ) ( )( ( )) w  (m1 + m2) = 1( ) 1( ( ( ))w  (m1 + m2) = 1( ) 1( )w ( m4 + m5) ≤ max { 1( ) 1( )w (m4), 1( ) 1( )w (m5)} = max { 1( ) 1( ( ( ))w  (m1), 1( ) 1( ( ( ))w  (m2)} = max{ 1 1( ) ( )( ( )) w   (m1), 1 1( ) ( )( ( )) w   (m2)} 4. 1 1( ) ( )( ( )) w   ( m1𝛼m2𝜃m3) = 1( ) 1( ( ( ))w  ( m1𝛼m2𝜃m3) = 1( ) 1( )w ( m4𝛼m5𝜃m6) ≥ min { 1( ) 1( )w (m4), 1( ) 1( )w (m5), 1( ) 1( )w (m6)} = min { 1( ) 1( ( ( ))w  (m1), 1( ) 1( ( ( ))w  (m2), 1( ) 1( ( ( ))w  (m3)} = min { 1 1( ) ( )( ( )) w   (m1), 1 1( ) ( )( ( )) w   (m2), 1 1( ) ( )( ( )) w   (m2)} 5. 1 1( ) ( )( ( )) w  ( m1𝛼m2𝜃m3) = 1( ) 1( ( ( ))w  ( m1𝛼m2𝜃m3) = 1( ) 1( )w ( m4𝛼m5𝜃m6) ≤ max { 1( ) 1( )w (m4), 1( ) 1( )w (m5), 1( ) 1( )w (m6)} = max { 1( ) 1( ( ( ))w  (m1), 1( ) 1( ( ( ))w  (m2), 1( ) 1( ( ( ))w  (m3)} Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 450 https://internationalpubls.com = max{ 1 1( ) ( )( ( )) w  (m1), 1 1( ) ( )( ( )) w  (m2), 1 1( ) ( )( ( )) w  (m3)} 6. 1 1( ) ( )( ( )) w  ( m1𝛼m2𝜃m3) = 1( ) 1( ( ( ))w  ( m1𝛼m2𝜃m3) = 1( ) 1( )w ( m4𝛼m5𝜃m6) ≤ max { 1( ) 1( )w (m4), 1( ) 1( )w (m5), 1( ) 1( )w (m6)} = max { 1( ) 1( ( ( ))w  (m1), 1( ) 1( ( ( ))w  (m2), 1( ) 1( ( ( ))w  (m6)} = max{ 1 1( ) ( )( ( )) w   (m1), 1 1( ) ( )( ( )) w   (m2), 1 1( ) ( )( ( )) w   (m3)} Then 2 ( ( ))w  is a “tripolar fuzzy ternary Γ−sub semiring” of M2. Hence (𝛘(𝛗1), M2, 𝚪) is a “tripolar fuzzy soft ternary Γ−semiring” over M2. Theorem 3.11: If M1 and M2 are TGSRs, 𝛘 : M1 → M2 is a TGSRH, (𝛗1, W1, 𝚪) and (𝛗2, W2, 𝚪) are TFSTGSR over M1 and (𝛗1, W1, 𝚪) is a “tripolar fuzzy soft ternary Γ−sub semiring” of (𝛗2, W2, 𝚪). Then (𝛘(𝛗1), W1, 𝚪) and (𝛘(𝛗2), W2, 𝚪) are “tripolar fuzzy soft ternary Γ−sub semirings” over M2 and (𝛘(𝛗1), W1, 𝚪) is a “tripolar fuzzy soft ternary Γ−sub semiring” of (𝛘(𝛗2), W2, 𝚪). Proof: Since 11( ( ))w  = 1 1( ( ( ))w  is a tripolar fuzzy ternary Γ-sub semiring of M2 ∀ w1 ∈ W1 and 22( ( ))w  = 2 2( ( ( ))w  is a tripolar fuzzy ternary Γ−sub semiring of M2 ∀ w2 ∈ W2. Hence (𝛘(𝛗1), W1, 𝚪) and (𝛘(𝛗2), W2, 𝚪) are tripolar fuzzy soft ternary Γ−semiring over M2. Since (𝛗1, W1, 𝚪) is a tripolar fuzzy soft ternary Γ−sub semiring of (𝛗2, W2, 𝚪). 11( )w is a tripolar fuzzy ternary sub semiring of 2 2( )w . Hence 1 1( ( ( ))w  is a tripolar fuzzy ternary Γ−sub semiring of 2 2( ( ( ))w  ∀ w1 ∈ W1. Therefore (𝛘(𝛗1), W1, 𝚪) is a tripolar fuzzy soft ternary Γ−sub semiring of (𝛘(𝛗2), W2, 𝚪). Theorem 3.12: If (𝛗1, W1, 𝚪) and (𝛗2, W2, 𝚪) are TFSTGSRs over M1 and M2 respectively and (𝛘, ψ) is a TFSH from (𝛗1, W1, 𝚪) onto (𝛗2, W2, 𝚪). then the pre-image of (𝛗2, W2, 𝚪) under TFSTGSRH is a “tripolar fuzzy soft ternary Γ−sub semiring” of (𝛗1, W1, 𝚪) over M1. Proof: By definition 3.4., (χ, ψ) -1 (φ2, W2, Γ) = 1 1 2 2( ( ), ( ))W    . Define 1 1 2 1( ( )) ( )w m  = ( )1 2 1( ( )) w m    ∀ m1 ∈M1 & w1 ∈ 1 2( )W  . If m1, m2, m3 ∈M1 & 𝛼, 𝜃 ∈Γ. Then 1. 2 1 1 1 2( ( )) ( )w m m   = 2 1( ) 1 2( ( ))w m m    = 2 1( ) 1 2( ( ) ( ))w m m    ≥ min{ 2 1 2 1( ) 1 ( ) 2( ( )), ( ( ))w wm m       } = min { 2 1( )  (m1), 2 1( )  (m2)} 2. 2 1 1 1 2( ( )) ( )w m m   = 2 1( ) 1 2( ( ))w m m    = 2 1( ) 1 2( ( ) ( ))w m m    ≤ max{ 2 1 2 1( ) 1 ( ) 2( ( )), ( ( ))w wm m       } = max { 2 1( )  (m1), 2 1( )  (m2)} 3. 2 1 1 1 2( ( )) ( )w m m   = 2 1( ) 1 2( ( ))w m m    = 2 1( ) 1 2( ( ) ( ))w m m    ≤ max{ 2 1 2 1( ) 1 ( ) 2( ( )), ( ( ))w wm m       } = max { 2 1( )  (m1), 2 1( )  (m2)} Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 451 https://internationalpubls.com 4. 2 1 2 1 1 3( ( )) ( )w m m m    = 2 1 ) 2 3( 1( ( ))w m m m     = 2 1( ) 1 2 3( ( ) ( ) ( ))w m m m     ≥ min{ 2 1 2 1 2 1( ) 1 ( ) 2 ( ) 3( ( )), ( ( ), ( ( ))w w wm m m           } = min { 2 1( )  (m1), 2 1( )  (m2) 2 1( )  (m3)} 5. 2 1 2 1 1 3( ( )) ( )w m m m    = 2 1 ) 2 3( 1( ( ))w m m m     = 2 1( ) 1 2 3( ( ) ( ) ( ))w m m m     ≤ max{ 2 1 2 1 2 1( ) 1 ( ) 2 ( ) 3( ( )), ( ( ), ( ( ))w w wm m m           } = max { 2 1( )  (m1), 2 1( )  (m2) 2 1( )  (m3)} 6. 2 1 2 1 1 3( ( )) ( )w m m m    = 2 1 ) 2 3( 1( ( ))w m m m     = 2 1( ) 1 2 3( ( ) ( ) ( ))w m m m     ≤ max{ 2 1 2 1 2 1( ) 1 ( ) 2 ( ) 3( ( )), ( ( ), ( ( ))w w wm m m           } = max { 2 1( )  (m1), 2 1( )  (m2) 2 1( )  (m3)} Thus 1 1 2( ( ))w  is a “tripolar fuzzy ternary Γ−sub semiring” of M1 ∀ w1 ∈ 1 2( )W  . ∴ 1 1 2 2( ( ), ( ))W    is a “tripolar fuzzy soft ternary Γ−sub semiring” of (𝛗1, W1, 𝚪) over M1. 4. 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